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Introduction to Non-Standard Analysis

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Introduction to Non-Standard Analysis INTRODUCTION TO NON-STANDARD ANALYSIS DIEGO ANDRES BEJARANO RAYO Abstract. In this paper we introduce the basic concepts of non-standard analysis. First, we provide an overview of first order logic and the theory of filters to successfully construct the set of hyperreal numbers, which we will use as our object of study. Using the transfer principle, a simple corollary ofLo´s'sTheorem, we not only introduce the non-standard notions of continuity, differentiability and Riemann integrability for functions of one variable, but also show that they are equivalent to the standard concepts taught in a first-year calculus course. Contents 1. First Order Logic. 1 2. Filters, Ultrafilters, Ultraproducts and Ultrapowers 3 3. The Structure of the Ultraproduct 6 4. The Structure of the Hyperreals 7 5. Continuity and Differentiability 11 6. Integration 14 7. Conclusion 17 Acknowledgments 17 References 17 1. First Order Logic. Before making an introductory exposition of the ideas behind non-standard analysis and the related construction, it is important that the reader familiarizes himself with the theory of first order logic. This section presents an introduction to these ideas, with a few examples. The system of first order logic is defined inductively, starting by defining the alphabet or the collection of symbols that will be used.1 Definition 1.1. The alphabet of first order logic is a set containing the following elements: • An infinite list of constant symbols: a; b; c; a1; b1; c1;::: • An infinite list of variable symbols: x; y; z; x1; y1; z1;::: • An infinite list of function symbols: f; g; h; f1; g1; h1;::: • An infinite list of relation symbols: P; R; Q; P1;R1;Q1;::: • Logical connectors: ^; _; :; !; $, with their usual interpretation. • Logical quantifiers: 8; 9. • The equality symbol: =. • The two parentheses \(" and \)". Note that every function and relation symbol is an n-placed function or relation symbol. This number is commonly referred to as the arity of the function or relation. For example, a function with arity 1 is called unary, with arity 2 is called binary, so on and so forth. We will denote the arity of a function as ar(f), so that for a binary function f, for example, we have ar(f) = 2. Definition 1.2. A term is a string of symbols from the alphabet that is defined recursively as follows: 1The material presented in this section is based off of [1] and [2]. 1 (1) Every constant is a term. (2) Every variable is a term. (3) If f is a function with arity n and t1; : : : ; tn are terms then f(t1; : : : ; tn) is also a term. (4) A string of symbols is a term if it can be constructed applying the previous steps finitely many times. At this point, it becomes important to point out that only the ^; : and 9 logical symbols are worth mentioning, since all the other logical connectors can be derived from these three. Note that ( _ ') = :((: ) ^ (:')), ( ! ') = :( ^ (:')), ( $ ') = :( ^ (:')) ^ :(' ^ (: )) and (8x)' = :(9x:'). Therefore, from now on, we will only mention the three necessary connectors, and assume that the rest follow from these equalities. Definition 1.3. A formula is a string of symbols from the alphabet that is defined recursively as follows: (1) If t1 and t2 are terms then (t1 = t2) is a formula. (2) If R is a relation with arity n and t1; : : : ; tn are terms then (R(t1; : : : ; tn)) is a formula. (3) If ' is a formula then so is :'. (4) If ' and are formulas, then so is (' ^ ). (5) If ' is a formula and x is a variable then (9x)' is also a formula. (6) A string of symbols is a formula if it can be constructed by finitely many applications of the previous steps. This also implies that, for example, ( _ ') is a formula if, and only if, and ' are formulas by using the fact that ( _ ') = :((: ) ^ (:')) and the definition above. Remark 1.4. The formulas described in 1.3.1 and 1.3.2 are called atomic formulas. Also, note that (1) can be understood as a binary relation R=(t1; t2) which is true if, and only if, t1 = t2. One of the limitations of first order logic is that it does not allow quantification over relations, only vari- ables. This is the key difference between first and higher order logic. While the definitions in this section will focus on developing first order logic, this distinction between first order and higher logic statements will be key in the development of non-standard analysis. Next, we will discuss the difference between free and bound variables. In not so rigorous terms, we say that a variable is free if it doesn't appear next to a quantifier and bound otherwise. Definition 1.5. Let ' be a formula. We define the set of free variables of ', denoted as FV ('), inductively as follow: (1) If ' = (t1 = t2), then FV ( ) = fx j x appears in t1 or t2g (2) If ' = (R(t1; : : : ; tn)) for some relation of arity n, then FV (') = fx j x appears in ti for some 1 ≤ i ≤ ng. (3) If ' = (: ), where is a formula, then FV (') = FV ( ). (4) If ' = (µ ^ ν), where µ and ν are formulas, then FV (') = FV (µ) [ FV (ν). (5) If ' = (9x) , where is a formula, then FV (') = FV ( ) n fxg. Definition 1.6. A formula ' is called a sentence if it has no free variables, meaning FV (') = Ø. Definition 1.7. A language L is a set containing all logical symbols and quantifiers (including the equality sign and the parenthesis) and some arbitrary number of constants, variables, function symbols and relation symbols. It is understood that all formulas made from any language L follow the previous rules. Definition 1.8. Let A be some nonempty set and V ⊂ L be the set of all variables in L.A variable assignment is a mapping β : V ! A, which assigns elements of A to all variables in V . Particularly, for some element k 2 A, some variable x 2 V and some variable assignment function β, there is a function β[x; v] defined as ( k if x = y β[x; k](y) = β(y) if x 6= y 2 Definition 1.9. A model or structure M for some language L is an ordered triple M = (A; I; β), where A is a nonempty set, β is a variable assignment function and I is an interpretation function with domain the set of all constants, relations and function symbols in L such that: (1) For every constant symbol c 2 L, we have that I(c) 2 A. (2) For every function symbol f 2 L with arity n, we have that I(f) 2 An × A. Meaning I(f) is a function of arity n defined on A. (3) For every relation symbol R 2 L with arity n, we have that I(R) ⊂ An. Meaning I(R) is the set of all n-tuples that satisfy R under I. A is frequently called the universe of M. Note that most authors define models as the pair (A; I) and leave the variable assignment function out, even when some have defined such a function. The reason why we decide to define models as a triple is simple; it makes the following definition easy to understand and does not cause any real change on what we consider a model to be. Definition 1.10. Let L be a language and M = (A; I; β) a model of L. Then the interpretation of any term t, denoted as (t)I,β, of symbols in L is defined as follows: • If t = c for some constant c, then (t)I,β = I(c). • If t = x for some variable x, then (t)I,β = β(x). I,β I,β I,β • if t = f(t1; : : : ; tn) for some function f of arity n, then (t) = I(f)((t1) ;:::; (tn) ). Definition 1.11. Let L be a language, M = (A; I; β) a model for L and ' some formula in L. Then, we say that M satisfies ' and write M j= ' or (A; I; β) j= ' whenever: • If ' = R(t1; : : : ; tn) for some relation R 2 L of arity n, meaning ' is atomic, then (A; I; β) j= ' if I,β I,β ((t1) ;:::; (tn) ) 2 I(R). • If ' = : for some atomic formula , then (A; I; β) j= ' if (A; I; β) does not satisfy . • If ' = (µ^ν) for some atomic formulas µ and ν, then (A; I; β) j= ' if (A; I; β) j= µ and (A; I; β) j= ν. • If ' = (9x) for some atomic formula , then (A; I; β) j= ' if there exists some k 2 A such that (A; I; β[x; k]) j= . Note that in this case we assume that x is a free variable on . This concludes the exposition of first order logic. While the reader might feel like the next section is quite disconnected with the previous one, both are essential in constructing the basic framework of non-standard analysis. 2. Filters, Ultrafilters, Ultraproducts and Ultrapowers Filters are a way of formalizing the notion of \big" within set theory. The following definitions set the groundwork for the study of filters, ultrafilters, ultraproducts, ultrapowers and their application within logic.2 Definition 2.1. A filter F on a set I is a set F ⊂ P(I) such that i. I 2 F ii. If X 2 F and X ⊂ Y , then Y 2 F for all X; Y 2 P(I). iii. If X; Y 2 F , then X \ Y 2 F for all X; Y 2 P(I).
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